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Fundamentals of

Albrecht Jander Oregon State University

Part I: H, M, B, χ, µ Part II: M(H) Magnetic from Current in a Wire

H dl

I r R c dl

Ampere’s Circuital Law: Biot-Savart Law:

× Jean-Baptiste Biot = = (1774 –1862) c 퐼4�푙⃗ 푅� c � 퐻 ∙ 푑푙⃗ 퐼 푑퐻 2 휋 푅 = [A/m] 퐼 André-Marie Ampère 2휋휋 Félix Savart (1775–1836) 퐻 (1791 – 1841) from Current Loop

H

r I

Use Biot-Savart Law: In the center: × = = [A/m] 퐼4�푙⃗ 푅� 푑퐻 2 퐼 휋 푅 퐻 2� Magnetic Field in a Long

H I JS I N turns

L

Inside, field is uniform with:

= [A/m] = [A/m] 푁� 푆 퐻 퐿 퐻 퐽 Magnetic Field from “Small” Loop ()

H

r I Magnetic Field from Small Loop (Dipole)

H

R θ Compare to Electric dipole:

For R >> r

= 2 cos + sin ( ) [A/m] 4 퐼퐼 퐻 3 풓� 휃 휽� 휃 Magnetic (dipole)휋 :푅 = [Am2]

푚 퐼�⃗ Magnetic Field of a Dipole

H

R θ

[A·m] For R >> r 푞푚 = 2 cos + sin ( ) [A/m] 4 푞푚푑 Magnetic퐻 (dipole) moment:3 푟̂ 휃 휃� 휃 휋푅 = [Am2] 푚 푞푚푑⃗ Magnetic Poles and

Electric charge Magnetic charge, or “pole”

+ q + qm

= 4 푞 [A·m] 퐷 2 휋푅 Magnetic field from a magnetic “monopole” 푞푚

= [A/m] 4 푞푚 퐻 2 휋푅 Magnetic Field of a Dipole

+ qm d

̶ qm

[A·m]

푞푚 Magnetic Field of a Dipole

H

R θ

[A·m] For R >> r 푞푚 = 2 cos + sin ( ) [A/m] 4 푞푚푑 Magnetic퐻 (dipole) moment:3 푟̂ 휃 휃� 휃 휋푅 = [Am2] 푚 푞푚푑⃗ Orbital

orbiting a nucleus are like a circulating current producing e- a magnetic field.

•For an with charge qe orbiting at a radius R with f, the Orbital Magnetic Moment is 2 2 m = IA = −qe f πR [Am ] • It also has an Orbital Angular

L = mevR = me 2πRf R • Note: m − q = e = L 2me Orbital Moment is Quantized

Bohr model of the

λdB •DeBroglie is:

e- h R λdB = mev

: orbit must be integer number of h h 2πR = NλdB = N = N

mev me 2πRf • Thus the orbital magnetic moment is quantized:

2 h qe qe m = −qe f πR = N = N

2π 2me 2me • Magnetic moment restricted to multiples of

Bohr Magneton

qe −24 2 µB = = 9.2742×10 [Am ] (1892-1987) 2me

• Spin is a property of subatomic particles (just like charge or mass.) • A particle with spin has a moment and . Pauli and Bohr contemplate • Spin may be thought of as conceptually the “spin” of a tippy-top arising from a spinning sphere of charge. (However, note that also have spin but no charge!) Electron Spin

• When measured in a particular direction, the measured angular momentum of an electron is  L = ± z 2

• When measured in a particular direction, the measured magnetic moment of an electron is Compare: qe mz = sz = ±µB Spin Orbital me mz = ±µB mz = NµB • We say “spin up” and “spin down” m − q m − q = e = e • Note: m − q L me L 2me = e L me m − q g = 2 = g e g =1 L 2me Stern-Gerlach Experiment - 1922

Demonstrated that magnetic moment is quantized with ±µB

Otto Stern (1888-1969)

Walther Gerlach, (1922). "Das magnetische Moment des Walter Gerlach Silberatoms". Zeitschrift für Physik A Hadrons and Nuclei 9. (1889-1979) Stern-Gerlach Experiment Numbers of Electron Orbitals

•Electrons bound to a nucleus move in orbits identified by quantum numbers (from solution of Schrödinger's equation):

n 1… principal , identifies the “shell” l 0..n-1 angular momentum q. #, type of orbital, e.g. s,p,d,f

ml -l..l ms +½ or -½

For example, the electron orbiting the Spectroscopic notation nucleus (in the ) has: l=0 s l=1 p n=1, l=0, m =0, m =+½ l s l=2 d l=3 f

In spectroscopic notation: 1s1 n l # of electrons in orbital 1s22s2

Spin and Orbital Magnetic Moment

• Total orbital magnetic moment (sum over all electron orbitals)

mtot _ orbital = µB ∑ ml • Total

mtot _ spin = 2µB ∑ ms

full orbitals: no net moment E.g. : 1s22s22p63s23p63d64s2 How to fill remaining d orbitals (l=2) 6 electrons for 10 spots:

ml=-2 ml=-1 ml=0 ml=+1 ml=+2 Hund’s rules: ms=+½      Maximize Σms ms=-½ 

Then maximize Σml m = 2µ m = 4µ tot _ orbital B tot _ spin B

Magnetization, M

Atomic magnetic moment: What is H here? [Am2] M Object푚 magnetic푎푎𝑎 moment:

2 . = [Am ]

Magnetic moment per unit volume: Atomic푚표표표 :∑ 푚푎푎𝑎 . [1/m3] = [A/m] = 푁 푚표표표 푑 푀 ≝ 푚푎푎𝑎푑 푣푣� 푣푣� Equivalent Loop Currents

Atomic magnetic moment: 퐼 What is H here? [Am2] = I M

Object푚푎푎 magnetic𝑎 moment:�⃗

2 . = [Am ]

Magnetic moment per unit volume: Atomic푚표표표 density:∑ 푚푎푎𝑎 . [1/m3] = [A/m] = 푁 푚표표표 푑 푀 ≝ 푚푎푎𝑎푑 푣푣� 푣푣� Equivalent Surface Current

Atomic magnetic moment: 퐼 What is H here? = I [Am2] M 푛� [A/m] 푎푎𝑎 J Object푚 magnetic moment:�⃗ S

2 . = [Am ]

Equivalent surface current: Atomic푚표표표 density:∑ 푚푎푎𝑎 = × [A/m] [1/m3] = 푁 퐽⃗푆 푀 푛� 푑 푣푣� Magnetic Pole Model

Atomic magnetic moment: What is H here?

2 = [Am ] M

Object푚푎푎 magnetic𝑎 푞 푚moment:푑⃗

2 . = [Am ] Magnetic moment per unit volume: Atomic푚표표표 density:∑ 푚푎푎𝑎 [A/m] = [1/m3] 푚 푁 푀 ≝ 푑 푣푣� 푣푣� Equivalent Surface Pole Density

Surface :

[Am] = [m2] 푞푚 Atomic magnetic moment: 휌푚푚 2 푎�푎푎 = [Am ] M

Object푚푎푎 magnetic𝑎 푞 푚moment:푑⃗

2 . = [Am ]

Magnetic pole density: Atomic푚표표표 density:∑ 푚푎푎𝑎 = [A/m] [1/m3] = 푁 휌푚푚 푀 ∙ 푛� 푑 푣푣� Example: NdFeB Cylinder 3 cm A 2 cm M JS

6 M =10 [A/m] 6 [A/m] JS =10

vol 14·10-6 [m3] I = 20000 [A] -4 [m2] 2 A 7·10 m =≃ 14 [Am ] m = IA = 14 [Am2] ≃ Example: NdFeB Cylinder 3 cm + + ++ + + ρ + + ++ + + + ms 2 cm M

6 [A/m] 6 M =10 ρms =10 [A/m]

vol 14·10-6 [m3] A 7·10-4 [m2] [Am] 2 q = 700 m =≃ 14 [Am ] m≃ d =2·10-2 [m] [Am2] m = qmd = 14 Example: NdFeB Cylinder

+ + ++ + + + + ++ + + + M

Result is the same external to magnetic material.

Hint: we are “far” away from the dipoles!!! Example: NdFeB Cylinder

+ + ++ + + + + ++ + + + M

Result is different inside the magnetic material. The Constitutive Relation

= ( + )

푩 흁ퟎ 푯 푴 B = Magnetic density [ ] H = Magnetic field, “Magnetizing ” [A/m] M = [A/m] 2 µ0 = Magnetic constant, “Permeability of free space” [Tesla-m/A] [/m] [N/A ]

Note, in , M is zero:

=

푩 흁ퟎ푯 What is µo ?

µ0 comes from the SI definition of the : The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to 2×10−7 per meter of length.

I 1 = = I 2 흁ퟎ푰ퟏ 푩 흁ퟎ푯 F = × ퟐ흅� B 푭 푰ퟐ푳 푩 ’s Equations ()

Differential form form

= =

훁 ∙ 푩 ퟎ � 푩 ∙ 풅풔 ퟎ × = =

훁 푯 푱⃗ � 푯 ∙ 풅풍⃗ 푰 With:

= ( + )

ퟎ 푩 흁= 푯 ( 푴 + ) = = [A/m2]

훁 ∙ 푩 흁ퟎ 훁 ∙ 푯 훁 ∙ 푴 훁 ∙ 푯 −훁 ∙ 푴 흆풎 Magnetic charge density i.e. magnetic charge/volume Example: NdFeB Cylinder

+ + ++ + + + + ++ + + +

M B H

= ( + )

푩 흁ퟎ 푯 푴 Boundary Conditions

B1┴ B2┴ H1|| H2||

The perpendicular component of B The transverse component of H is is continuous across a boundary continuous across a boundary (unless there is a true current on the boundary) Demagnetizing Fields

+ + ++ + + + + ++ + + +

M Hd

푯풅 ∝ 푴 =

푯풅 −푵� Demagnetizing Factor Demagnetizing Factors: Special Cases = + + =

푯풅 −푵� 푵풙 푵풚 푵풛 ퟏ Sphere Thin Sheet

Long Rod

= = = = = ퟏ = 푵풙 푵풚 푵풛 푵풙 푵풚 ퟎ z ퟑ = = 푵풛 ퟏ x ퟏ 푵풙 푵=풚 y ퟐ 푵풛 ퟎ Demagnetizing Factors

“Demag” in ellipsoids of revolution are uniform.

=

푯풅 + −푵+� = 푵풙 푵풚 푵풛 ퟏ Example: Ba-Ferrite Sphere

Barrium Ferrite: ~ [kA/m]

푴 ퟏퟏ�

M H B

= = = = ( + ) 풙 풚 풛 ퟏ 푵 푵 푵 푩 = 흁ퟎ(푯 푴[kA/m] ) = ퟑ ퟎ 풅 푩 = 흁. ퟏ𝟏 [Tesla] 푯 =−푵� [kA/m]

푯풅 −𝟓 푩 ퟎ ퟏ�ퟏ Example: Ba-Ferrite Ellipsoid

Barrium Ferrite: ~ [kA/m]

푴 ퟏퟏ�

M H B = = . = ( + ) 풙 풚 푵 푵 ퟎ ퟒ� ퟎ = 푩 = 흁 (푯 . 푴 [kA/m] ) ퟎ 풅 푩 = 흁. ퟖ� �[Tesla] 푯 =−푵�. [kA/m]

푯풅 −�ퟔ � 푩 ퟎ ퟏ𝟏 Example: Ba-Ferrite Ellipsoid

Barrium Ferrite: ~ [kA/m]

푴 ퟏퟏ�

M H B = . = ( + ) 풛 푵 ퟎ ퟏ ퟎ = 푩 = 흁 (푯 푴 [kA/m] ) ퟎ 풅 푩 = 흁. ퟏ�ퟏ [Tesla] 푯 =−푵� [kA/m]

푯풅 −ퟏퟏ 푩 ퟎ ퟏ� Susceptibility and Permeability Ideal, linear soft magnetic material Ideal permanent M M

H H

= Real, useful magnetic material Susceptibility [unitless] M 푴 흌� = ( + )

푩 = 흁ퟎ(푯+ 흌)푯 H 푩 = 흁ퟎ ퟏ 흌 푯 Rel. Permeability 푩 흁ퟎ흁풓푯 [unitless] Example, Long Rod in a Long Solenoid

I =1 A = 1000 = 999 N/L = 1000/m 휇� 휒

= = 1 [kA/m] 푁� = = = 퐻 퐿 999 [kA/m] 푁� 푀= 휒� +휒 퐿 = (1000kA/m)= 1.25 [Tesla] or: 0 퐵 휇0 퐻 푀 휇 = = 1.25 [Tesla] Note: without the rod: = = 0.00125 [Tesla] 퐵 휇0휇�퐻 퐵 휇0퐻 Example, Soft Magnetic Ball in a Field

Hext

M Hd B

= = + =

푴 흌� 흌 푯풆�풆 푯풅 흌 푯풆�풆 − 푵� = = < ! + + 흌 흌 푴 푯풆�풆 흌풆풆풆 ퟑ‼ ퟏ 푵흌 ퟏ 푵흌 Example, Soft Magnetic Ball in a Field

B = = + =

푴 흌� 흌 푯풆�풆 푯풅 흌 푯풆�풆 − 푵� = = < ! + + 흌 흌 푴 푯풆�풆 흌풆풆풆 ퟑ‼ ퟏ 푵흌 ퟏ 푵흌 Example, Thin Film in a Field

Hext

= = + =

푴 흌푯풊풊�풊𝒊 흌 푯풆�풆 푯풅 흌 푯풆�풆 − 푴 = + 흌 푴 푯풆�풆 ퟏ 흌 = = + ퟏ 푯풊풊�풊𝒊 푯풆�풆 푩풊풊�풊𝒊 퐵Why?풆�풆 ퟏ 흌 and Zeeman F + + ++ + + + + ++ + + + Force: = B M Torque: 퐹⃗ 푞푚퐵 = × F Energy: 풯 푚 퐵 =

퐸 −푚 ∙ 퐵 Self Energy and Shape Anisotropy

M = Hd M 1 Hd 2 0 퐸 � 휇 푀 ∙ 퐻푑� Higher energy Lower energy 푣표표 “Shape Anisotropy”

= ( ) E 2 Hard Hard 1 2 퐸 2휇0푀� 푁푥 − 푁푌 푠푠� 휃

Easy Easy M Easy axis θ 0 180 360 Hard axis Magnetic Circuits

path length, l I = area,A N turns � 푯 ∙ 풅풍⃗ 푵� Assuming flux is uniform and contained inside permeable material:

= = 흓 = 푩 흁ퟎ흁풓푯 푨 -motive force, “MMF”푵� 흓흓 Reluctance, = 풍 Flux � 흁� Magnetic Circuits

�ퟏ �ퟐ

NI

ퟑ 품품품 � �품품품 흓

�ퟒ ��

Solve using electric circuit theory. Units Confusion

CGS Units SI Units

B [] B : [Tesla] H [] H : [A/m] M [emu/cc] M : [A/m] 4πM [Gauss]

= + = + ퟎ 푩 푯 ퟒ흅� 푩 흁 푯3 푴 [dyne/cm3] [J/m ]

푩 ∙ 푯 푩 ∙ 푯 + + = + + =

푵풙 푵풚 푵풛 ퟒ흅 푵풙 푵풚 푵풛 ퟏ Books on Magnetism

B.D. Cullity, Introduction to Magnetic Materials, Wiley-IEEE Press, 2010. (revised version with C.D. Graham). M. Coey, Magnetism and Magnetic Materials, Cambridge University Press, 2010. S. Chikazumi, Physics of Magnetism, John Wiley and Sons, 1984. R.C. O’Handley, Modern Magnetic Materials, John Wiley & Sons, 2000. R.L. Comstock, Introduction to Magnetism and Magnetic Recording, John Wiley & Sons, 1999. D. Jiles, Introduction to Magnetism and Magnetic Materials, CRC Press, 1998. Bozorth, ,1951 (reprinted by IEEE Press, 1993.)